An Eruv Tavshilin is a halachic device which allows one to cook food on the Festival for use on the Shabbat that immediately follows.
To allow one to cook on the Festival for use on Shabbat, one must prepare an eruv tavshilin which serves as a reminder that the cooking done on the Festival is for Shabbat only.
This eruv consists of a matzah and something cooked, for example, a boiled egg or a piece of fish, and is prepared before the onset of the Festival.
It symbolically serves as the beginning of the preparation of food for Shabbat. Thus, any subsequent cooking done on the Festival is considered to be a continuation of the preparation begun before the Festival.
If the last day of the Festival falls on a Shabbat, one must prepare an eruv tavshilin on Thursday afternoon before the festival begins so that one can cook on the Festival for Shabbat.
How to make an Eruv Tavshilin
Take a plate with one whole matzah and either a piece of cooked fish, cooked meat or a hard-boiled egg on it, (it is a good idea to wrap the matzah/challah and fish, meat or egg in aluminum foil to easily keep them apart from the rest of the foods in your home). Recite the following blessing:
BA-RUCH A-TAH ADO-NAI E-LO-HE-NU ME-LECH HA-OLAM ASHER KID-E-SHA-NU BE-MITZ-VO-TAV VETZI-VA-NU AL MITZVAT ERUV.
Then recite: “By virtue of this Eruv, we (the members of this household), shall be permitted to cook, bake, keep food warm, carry, light candles and do all preparations on Yom Tov for Shabbat.”
Put the foods you used for the Eruv away, and eat them on Shabbat. NOTE: The Eruv Tavshilin allows you to cook on Friday only for Shabbat, it does not allow you to cook from one day of the holiday to the next (ie: Thursday for Friday). It is permissible to cook only from a pre-existent flame, one that was lit before the onset of the Holiday on Thursday before sunset.